
The conjecture holds for \(M=2,3\)  [1]}.

If \(N<4M-2\alpha (M-1)-3\) , then \(\mathcal {A}\)  is not injective [2]}; here, \(\alpha (M-1)\le \log _2 M\)  denotes the number of 1's in the binary expansion of \(M-1\) .

For each \(M\ge 2\) , there exists an ensemble \(\Phi \)  of \(N=4M-4\)  measurement vectors such that \(\mathcal {A}\)  is injective [3]} (see also Section 2 of this paper).

If \(N\ge 4M-2\) , then \(\mathcal {A}\)  is injective for generic \(\Phi \)  [4]}.

